We report JWST/NIRSpec spectra of three distant T-type brown dwarfs identified in the Ultradeep NIRSpec and NIRCam ObserVations before the Epoch of Reionization (UNCOVER) survey of the Abell 2744 lensing field. One source was previously reported as a candidate T dwarf on the basis of NIRCam photometry, while the two other sources were initially identified as candidate active galactic nuclei. Ultradeep, low-resolution 1-5 μm spectra confirm the presence of molecular features consistent with T dwarf atmospheres, and comparison to near-infrared spectral standards infers spectral classifications of sdT1, T6, and T8-T9. The spectrum of the warmest source, UNCOVER-BD-1, shows evidence of subsolar metallicity, and a fit to atmosphere models indicates Teff = 1300 K and [M/H]~-1.0, making this one of the few examples of a spectroscopically-confirmed T subdwarf. The spectrum of the coldest source, UNCOVER-BD-3, places it near the T/Y dwarf boundary with Teff = 550 K, and we identify features of CH_4, CO, H_2O, and likely PH_3 in the 3.5-5.0 μm band. Our analysis suggests that PH3 is favored over CO_2 in this source, a possible indicator of subsolar metallicity. We estimate distances of 0.9-4.5 kpc from the Galactic midplane, and population simulations indicate high probabilities of membership in the Galactic thick disk or halo. Our simulations also indicate that there may be up to 7 T dwarfs and 5-6 L dwarfs in the Abell 2744 field down to F444W = 30 AB mag, roughly half of which are halo members. These results highlight the utility of deep JWST/NIRSpec spectroscopy for identifying and characterizing the oldest metal-poor brown dwarfs in the Milky Way.